I'm playing Undertale in a true pacifist run and after I fought Omega Flowey I closed down my game. I opened it back up and I'm at the beginning of the Asgore fight with my pie and everything. Is this supposed to happen?
It sounds like you completed a 0 EXP neutral run. Flowey gives you a hint in his dialogue after you beat him, telling you to be better friends with some of the monsters. If you have 0 EXP at this point, you will be reverted to your last SAVE point, with some extra options unlocked (so you don't have to do the entire run again).
In order to complete a pacifist run, you also need to complete "date" events with Papyrus, Undyne, and Alphys. If you've already completed dates with Papyrus and Undyne (Alphys isn't unlocked on the first run), you can backtrack until you get a call from Undyne. From there, just do what they ask, and you should have completed everything you need for a pacifist run.
You'll know if you did it properly when you reach Asgore again, because it'll start a bit differently.